Abstract
Large-scale renewable energy integration decreases the system inertia and restricts frequency regulation. To maintain the frequency stability, allocating adequate frequency-support sources poses a critical challenge to planners. In this context, we propose a frequency-constrained coordination planning model of thermal units, wind farms, and battery energy storage systems (BESSs) to provide satisfactory frequency supports. Firstly, a modified multi-machine system frequency response (MSFR) model that accounts for the dynamic responses from both synchronous generators and grid-connected inverters is constructed with preset power-headroom. Secondly, the rate-of-change-of-frequency (ROCOF) and frequency response power are deduced to construct frequency constraints. A data-driven piecewise linearization (DDPWL) method based on hyperplane fitting and data classification is applied to linearize the highly nonlinear frequency response power. Thirdly, frequency constraints are inserted into our planning model, while the unit commitment based on the coordinated operation of the thermal-hydro-wind-BESS hybrid system is implemented. At last, the proposed model is applied to the IEEE RTS-79 test system. The results demonstrate the effectiveness of our co-planning model to keep the frequency stability.
AS serious energy crisis spreads in the world, renewable energy sources (RESs) are deemed as the best way to achieve the sustainable development [
Nowadays, massive frequency response modes are excavated to supply adequate system inertia and maintain frequency within the acceptable range. RES-based power plants are studied to obtain virtual inertia and primary frequency response (PFR) by integrating frequency control loops. In this respect, various control strategies such as deloading and overproduction have been developed to facilitate the provision of frequency support from variable speed wind turbines (VSWTs) [
In recent years, frequency-constrained problems mainly concentrate on optimal operation such as economic dispatch (ED) and unit commitment (UC). For instance, [
With respect to planning problems, the frequency stability has been included into the siting and sizing issues of BESS. For instance, the contribution of inertia and droop controls from BESS is analyzed in [
In this paper, we mainly focus on RES planning that installs wind farms to facilitate renewable consumption. RES planners should impose frequency limits to secure frequency stability and avoid system collapse. In practice, few existing studies and projects have considered the frequency stability when making RES planning decisions. Without such considerations, a system planner tends to overinvest in RES, while the total delivery of inertial and PFR may be reduced in real time, leading to serious frequency deviations or renewable production curtailment. Therefore, a frequency-constrained planning model of generation and BESS has been coordinated to simultaneously satisfy the frequency stability and power consumption requirements. In general, the main contributions are threefold.
1) A novel multi-machine system frequency response (MSFR) model that incorporates thermal units, hydro units, wind farms, and BESS is proposed to analyze the dynamic frequency trajectory. On this basis, the multi-machine aggregation method is utilized to aggregate the MSFR model into an equivalent single-generator by converting all control loops to a thermal form. Meanwhile, the available frequency response power and power-headroom constraints are both constructed to activate sufficient PFR.
2) Two frequency constraints, namely the limitation on ROCOF and frequency nadir, are deduced from aggregated MSFR (AMSFR) model. The frequency nadir is further transformed into a system frequency response power constraint (FRPC). Furthermore, a data-driven piecewise linearization (DDPWL) method based on hyperplane fitting and data classification is proposed to linearize the highly nonlinear FRPC by solving a mixed-integer second-order cone problem.
3) A frequency-constrained coordination model of generation expansion planning and energy storage installation (GEP&ESI) is presented to supply enough system inertia and activate PFR. The short-term UC is inserted into a long-term planning procedure considering renewable portfolio standards (RPSs) to obtain high-penetration renewable integration. Meanwhile, a hybrid thermal-hydro-wind-BESS operation model is proposed to promote renewable consumption, which constructs extra water constraints for cascading hydro units. Afterwards, linearization methods comprising the big-M method, reformulation linearization technique (RLT), and McCormick envelopes are adopted to deal with bilinear constraints, which ultimately forms a mixed-integer linear optimization problem. Primary frequency reserves are also confined to satisfy the power-headroom requirement in the proposed MSFR model.
The remainder of this paper is organized as follows. The MSFR model is introduced in Section II. A DDPWL method is proposed and discussed in Section III. Then the GEP&ESI model that incorporates frequency limitations is presented in Section IV. The case studies are conducted and explained in Section V. Section VI concludes this paper.
RES-based inverters are always used to supply both inertial and droop controls through virtual synchronous machine (VSM) technique. In brief, the virtual inertia mimics synchronous inertia and releases over-generated power within milliseconds [
The frequency response power from wind farms can be expressed as (1) in the time domain. Meanwhile, the time delay is inserted into the transfer function in the frequency domain, as shown in (2). Since the frequency response power depends on de-loading operation, the maximum amounts of PFR are limited by renewable curtailment in constraint (3), which relates to the droop constant and the maximum frequency deviation at the same time.
(1) |
(2) |
(3) |
where and are the frequency deviations in the time domain and frequency domain, respectively; and are the total frequency response power in the time domain and frequency domain, respectively; and are the forecasted output power and curtailment power for wind farm w at time t, respectively; is the maximum frequency deviation; , , and are the inertia constant, droop constant, and response time of wind farm w, respectively; and is the reserve coefficient, which can be used to regulate the preset power-headroom for PFR.
In this paper, Lithium-ion batteries are selected as BESS devices, which activate quick PFR within 40 ms and keep the duration time from few minutes to hours [
(4) |
where is the frequency response power of BESS e in the frequency domain; and , , and are the gain coefficient, droop constant, and response time of BESS e, respectively.
Since the droop control is adopted to supply PFR, the timely frequency response power should be limited within the preset power-headroom. Similarly, the power-headroom depicted in
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F001.jpg)
Fig. 1 Schematic diagram of PFR from BESS.
On this basis, constraints (5) and (6) are proposed to stipulate ranges of frequency response power with respect to discharging and charging states. Constraint (7) gives an expression of total PFR power. Only one of and would take effect at any particular time, which can be guaranteed by integrating binary indicators ( and ) and adding exclusive constraint (8). The stored energy after frequency responses should be subject to the permitted energy ranges in (9) and (10), which provides enough energy headroom to sustain frequency responses for the duration time of PFR . Note that (9) gives out a clear correlation of the stored energy between the normal operation and post-regulation. Here, parts ①-③ refer to the changeable energy due to the frequency regulation. Specifically, part ① relates to the condition that BESS discharges power in normal operation. BESS charges in both normal operation and frequency regulation with part ②. On the contrary, BESS charges in normal operation but transforms to the discharging state after frequency regulation with part ③. Constraint (10) restricts the exclusive state through binary variables and .
(5) |
(6) |
(7) |
(8) |
(9) |
(10) |
where and are the indicators of discharging and charging operation states, respectively; is the total PFR power; and are the stored energy before and after frequency responses, respectively; and are the permitted minimum and maximum stored energy, respectively; and and are the discharging and charging efficiencies of BESS e, respectively.
An innovative MSFR model comprising synthetical frequency responses from SGs (both thermal and hydro) as well as converters (both wind farm and BESS) is integrated to maintain frequency stability. This MSFR model can be illustrated by the block diagram in
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F002.jpg)
Fig. 2 Schematic diagram of MSFR model.
For thermal units, Fh and Tr are the fraction of power generated by the high-pressure turbine and the reheat time of thermal unit, respectively; and Kg and Rg are the mechanical power gain coefficient and the governor speed constant, respectively. For hydro units, frequency responses mainly come from the synchronous inertia, governors, and hydro turbines, where the core component is the governor [
As can be observed from
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F003.jpg)
Fig. 3 Schematic diagram of AMSFR model.
Furthermore, the power gain factor Km of each synchronous generator and converter can be defined as:
(11) |
where is the maximum response capacity of device i; is the system base power; and , , , and are the sets of thermal units, hydro units, wind farms, and BESS, respectively.
In this paper, we assume that the system base power equals the hourly demand so that the damping coefficient always keeps the same in any time interval. Thus, the equivalent system inertia can be calculated by (12), which relates to the individual inertia and response capacity at the same time.
(12) |
where is the equivalent system inertia at time t; is the inertia constant for generator g; is the maximum frequency response capacity of generator g; refers to demand b at time t; is the frequency-support state of generator g at time t; and and are the sets of SGs and power grid buses, respectively.
To derive the equivalent frequency parameters, the transfer functions of wind farms, BESS, and hydro units can be converted into (13)-(15), respectively, in sequence.
(13) |
(14) |
(15) |
where and are the equivalent droop coefficients of wind farm and BESS, respectively; , , and are the equivalent power fractions of wind farm, BESS, and hydro unit, respectively; and is the equivalent time constant of hydro unit.
Afterwards, the newly-defined droop and governor speed constants can be described as (16) and (17), respectively. Similarly, the occupation proportion is defined as (18), and all proportions should sum up to one. On this basis, we construct the linear formula of FH and TR in (19) and (20), respectively.
(16) |
(17) |
(18) |
(19) |
(20) |
where , , and are the frequency-support states of hydro unit, wind farm, and BESS at time t, respectively; is the maximum capacity of hydro unit d; and is the maximum response capacity of BESS e.
Based on the block diagram in
(21) |
(22) |
The analytical formulation of in the time domain can be given as (23), where the damping frequency and coefficients and are defined in (24). Then, a step signal is settled for the power deviation.
(23) |
(24) |
According to the frequency deviation (23), we implement its derivation to deduce the maximum frequency deviation and related nadir time . These two metrics expressed as (25) are directly decided by the equivalent frequency parameters of system. In addition, the ROCOF is employed to limit the maximum changing speed of frequency at the initial time of frequency regulation, which enables to avert unexpected generator tripping. The formulation of ROCOF is shown as (26) [
(25) |
(26) |
Furthermore, the frequency deviation and ROCOF restrictions are constructed as (27) and (28), respectively. In this vein, the ROCOF limitation equals the system inertia requirement, which is totally linear and can be directly put into the optimization model. Moreover, the frequency deviation is transformed into the power expression in constraint (29) to reflect the basic requirement of system frequency response power. However, the right part defined as is a highly nonlinear multi-variable function, which is decided by the equivalent frequency parameters and should be further linearized in the next section.
(27) |
(28) |
(29) |
where is the nominal frequency; and is the maximum permitted ROCOF.
In this section, a traditional piecewise linearization (PWL) method from [
1) Due to the low sensitivity of the frequency deviation to the reheat time constant (about 1%) [
2) With respective damping and droop gains usually strictly prescribed within narrow ranges by the system operator, the equivalent damping constant of the system can be settled as a constant [
On this basis, the decision variable set can be further reduced as . To form a linear correlation between and frequency-support states, fitting variables coming from (17) and (19) are employed instead. Thus, the improved PWL function can be given as:
(30) |
(31) |
where and refer to the fitting coefficients, which are optimized to eliminate fitting errors between the real value and the fitting value ; counts the number of adopted hyperplanes; and is the set of hyperplanes.
Note that the second-order terms are always optimized to derive the appropriate fitting coefficients based on the dataset . The traditional PWL problem can be constructed as:
(32) |
Since the problem (32) minimizes the squared errors in the data set, the frequency stability is possibly destroyed at some points. Meanwhile, the over-conservativeness takes place if a lower fitting value adopts and facilitates more frequency supports. Meanwhile, enough data points and hyperplanes should be inserted to guarantee accuracy so that solution obstacles always exist. To overcome the shortcomings above, [
In this paper, a modified DDPWL methodology is proposed to deal with the nonlinear FRPC. Firstly, the initial problem (30)-(32) can be transformed into the problem (33)-(35) by defining . It can be found that the inner problem can be totally eliminated through (34) and (35), which guarantees the same value as the maximum function. Moreover, a logistic constraint that limits one efficient hyperplane in each variable region can be constructed as (36). Furthermore, constraints (37) and (38) are novelly added without loss of generality if all data points are positive. These two constraints shrink the feasible regions and accelerate the solution procedure [
(33) |
(34) |
(35) |
(36) |
(37) |
(38) |
where is a binary variable, when the
As the nonlinear FRPC is a hard constraint in our planning problem, it is not necessary to force the linearized value as close to its real value as possible at all data points. In this vein, a data-driven classification idea from [
(39) |
(40) |
(41) |
(42) |
(43) |
(44) |
(34)-(38) | (45) |
After deriving all PWL functions, we should deal with the maximum problem among all hyperplanes. On this basis, the big-M method [
(46) |
This section proposes a frequency-constrained GEP&ESI model, which can satisfy the required demand increase and supply enough frequency response power simultaneously. Since unit states impact system inertia and PFR directly, the short-term UC considering RPSs is inserted into the long-term planning model, which schedules sufficient frequency-support sources and promotes renewable consumption at the same time. Specifically, the objective function of the proposed model is to minimize the total cost as formulated in (47). The total cost comprises the investment costs of thermal units, wind farms, and BESS, fuel cost, unit startup cost, wind curtailment cost, load curtailment cost, load shifting-in cost, and load shifting-out cost, respectively. In this paper, the curtailable and shiftable demands are deemed as two efficient ways to obtain demand responses (DRs).
(47) |
where , , and are the sets of candidate thermal units, time intervals, and stochastic scenarios, respectively; , , , , , , , and are the coefficients of annual investment costs of thermal units, wind farms, and BESS, costs of fuel consumption and unit startup, scheduling of wind curtailment, load interruption, and load shifting, respectively; , , , ,, , , , and are decision variables including binary construction indicators of candidate thermal units, wind farms, and BESS, power output of conventional units (both thermal and hydro units), binary indicators of unit startup, curtailable power of wind farm and interrupted load, and shifting-in and shifting-out power of response demand, respectively; and and are the possibility of scenario s and the total hours of the target planning year, respectively.
This problem should subject to the following constraints.
(48) |
where sets , , , and are the indices of synchronous units (both thermal and hydro units), wind farms, BESS connected to bus b, and all transmission lines, respectively; and and are transmission power flow and the load demand after DRs, respectively.
DC power flow for transmission lines can be expressed as (49). Also, the maximum power flow should be limited within the line capacity by constraint (50).
(49) |
(50) |
where and are the voltage phase angles of origin and receive buses for line l, respectively; and , , and are the susceptance, minimum and maximum power flows of line l, respectively.
Constraint (51) implements the logistic correlation between on/off states, startup, and shutdown actions. Constraint (52) declares the exclusive correlation between startup and shutdown actions. The minimum on/off time maintained for conventional units is limited by constraints (53) and (54), respectively. Moreover, constraint (55) imposes the limit on active output power. The ramp-up and ramp-down rates for conventional generators are restricted in (56).
(51) |
(52) |
(53) |
(54) |
(55) |
(56) |
where is the indicator of unit shutdown; is the indicator of on/off state of unit g; and are the maximum ramp-up and ramp-down power of unit g, respectively; and are the minimum on/off time of unit g, respectively; and are the duration hours of on/off states at the begging of typical days, respectively; and are the minimum and maximum output power offered by unit g, respectively; and and are the number of time intervals and the duration hour of adjacent time, respectively.
Nowadays, in order to reduce emissions and promote investment in renewable generation, new integration incentives such as RPSs, feed-in tariffs (FITs), and production tax credits have been implemented in more than 150 countries [
(57) |
(58) |
where represents the forecasted demand connected to bus b at time t in scenario s.
The operation constraints of BESS are listed in (59)-(65). In general, the charging and discharging processes of BESS can be described as (59). Constraint (60) represents the lower and upper bounds of SOC [
(59) |
(60) |
(61) |
(62) |
(63) |
(64) |
(65) |
(66) |
where and are the remaining energy of BESS e at the initial and end time on representative days, respectively.
For hydro units, the limitations of output power, minimum on/off time, and ramp-up/down power in normal operation can be similarly constructed as thermal units. Besides, the unique constraints for cascading hydro unit can be listed as (67)-(70) [
(67) |
(68) |
(69) |
(70) |
where and are the water discharge and reservoir volume of hydro unit d, respectively; is the indicator of on/off state; is the natural inflow to reservoir of hydro unit d; and are the minimum and maximum water discharges of hydro unit d, respectively; and are the minimum and maximum reservoir volumes of hydro unit d, respectively; and and are the initial and terminal reservoir volumes of hydro unit d, respectively.
The water-to-power conversion of cascaded hydro units is expressed by a head-dependent function in (71) [
(71) |
(72) |
In this paper, the curtailable and shiftable demands are employed to obtain DRs [
(73) |
(74) |
(75) |
(76) |
(77) |
Constraints (46) and (78) are utilized to guarantee enough frequency response power, while the constraint (79) requires enough system inertia to limit ROCOF. The equivalent frequency parameters can be calculated through (12), (17), and (19). Note that all equivalent frequency parameters just depend on the frequency-support states if settling a fixed unbalanced power. In this vein, the inner maximum problem can be eliminated by big-M method as introduced in Section III. Furthermore, constraints (80)-(83) preset enough power-headroom to guarantee PFR from thermal units, hydro units, wind farms, and BESS, respectively. Here, represents the maximum frequency deviation at the quasi-steady state, which is generally utilized to estimate the required power-headroom for PFR. If settling in (23), can be estimated by (84), which mainly correlates to the values of , , , and .
(78) |
(79) |
(80) |
(81) |
(82) |
(83) |
(84) |
Unfortunately, the bilinear constraints render the proposed model unable to be dealt with by the off-the-shelf solvers. These constraints can be classified into two categories, i.e., ① the multiplier of binary and continuous variables such as (5), (6), and (9); ② the multiplier of two continuous variables such as (72). With respect to the former category, auxiliary variables denoted as and are adopted to linearize constraints (5) and (6). The related auxiliary constraints are proposed as (89) and (90) through big-M method [
(89) |
(90) |
(91) |
(92) |
With respect to the bilinear problem with the multiplier of two continuous variables, the RLT has been used to deal with the nonlinear part in (72). This method is widely used to solve continuous factorable nonconvex optimization problems through tightly effective relaxations in a higher-dimensional space [
1) The nonlinear term must be expressed as the form of bilinear products.
2) All variables are bounded within the preset ranges. It can be observed from constraints (68), (69), and (72) that the nonlinear term entirely conforms to the above conditions.
On this basis, we reformulate the nonlinear terms through substituting by auxiliary, nonnegative, and continuous variables . Meanwhile, McCormick envelopes [
(93) |
(94) |
In this section, the proposed frequency-constrained GEP&ESI model is implemented on a modified IEEE RTS-79 test system [
The modified IEEE RTS-79 test system consists of 24 buses, 33 generation units, 38 lines, and 17 load points. The demand has enlarged to be 1.1 times the original value for highlighting the needs of generation expansion. The schematic diagram of this test system is depicted in
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F004.jpg)
Fig. 4 Schematic diagram of IEEE RTS-79 bus test system.
Note: U76 represents the thermal units with capacity of 76 MW.
The scenario technique is used to describe uncertainties through K-means clustering [
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F005.jpg)
Fig. 5 Power proportion on four typical days.
Besides, the minimum consumption rate and the maximum curtailment rate are set as 15% and 30%, respectively, in the following case studies except otherwise stipulated. In this paper, DRs can be implemented in buses 3, 10, 13, 15, and 18, where the maximum curtailable and shiftable percentages are settled as 10%. Meanwhile, 80 $/MWh and 5 $/MWh should be paid for such curtailable and shiftable loads, respectively [
In this subsection, the performance of the linearized FRPC compared with the original nonlinear constraint has been examined. Two types of errors are defined as: ① Type I, satisfying the FRPC according to the linearized value but actually violating the FRPC; ② Type II, violating the FRPC according to the linearized value but actually satisfying the FRPC. The averaged fitting error can be calculated as:
(95) |
where counts the number of error points; and are the calculated values of frequency response power through linearized FRPC and initial nonlinear formulation (29), respectively; and is the set of error points.
Furthermore, the proposed DDPWL method is compared with three other methods that are widely used in dealing with nonlinear frequency constraints. The performances of different methods are compared in terms of the linearization error, fitting data, computation time, and the total cost. It should be noted that the bound extraction is already introduced in Section III that enumerates to seek for the worst-case frequency parameters. Comparison results of four widely-used methods can be found in
In theory, no error exists for the bound extraction since the precise bound with related frequency parameters can be found by enumerating all possible data points. However, it is almost impossible to calculate the frequency response power with consideration of frequency-support states for all devices, e.g., more than
The above challenges can be solved by the proposed DDPWL method, which is based on the idea of data classification. It can be observed that the simulation results are promising with acceptable Type I error (0.7%) and negligible Type II error (0.02%), which almost have no impact on the total cost. Meanwhile, the computation time of DDPWL is much shorter than that of the other three methods because it only minimizes the error around the limit. In conclusion, the proposed DDPWL method can be utilized to construct the linearized FRPC with high accuracy and enhanced computation efficiency for the rest of the study.
In this sub-section, the planning results with/without frequency constraints (denoted as WFC and WOFC, respectively) are listed in
Note: 1(1,76) represents installing one thermal unit with capacity of 76 MW at bus 1; 1(300) represents installing one wind farm with capacity of 300 MW at bus 1. The scheme of BESS only lists the integration bus number.
Besides, more conventional units have to be started up and keep the minimum output power in WFC case, which increases the fuel cost from 253.1 M$ to 281.1 M$. In order to explore the effects of frequency constraints on operation schedules, the output power profiles of both existing and candidate units in all time intervals are depicted in
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F006.jpg)
Fig. 6 Power profiles in all time intervals in WFC and WOFC cases. (a) WFC. (b) WOFC.
The total frequency response power from SGs, wind farms, and BESS on four representative days is shown in
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F007.jpg)
Fig. 7 System frequency response power in WFC and WOFC cases.
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F008.jpg)
Fig. 8 Frequency trajectories of two cases in case of demand increase.
A serious frequency drop takes place in WOFC case and exceeds the permitted frequency limitation ultimately. From the perspective of ROCOF, the required inertia for this test system is for all time intervals. We can find that the minimum system inertia for WFC and WOFC cases are allocated with and , respectively, which addresses slight effects of ROCOF restriction in this test system. The above results emphasize the great significance of integrating frequency constraints into planning problems, especially for future environments with high penetration of renewable energy. It also powerfully demonstrates the effectiveness of our frequency-constrained GEP&ESI model in guaranteeing frequency stability.
In this subsection, the planning results for different response modes are compared to emphasize the superiority of our proposed MSFR model. Here, four response modes are declared as: ① only SGs; ② both SGs and wind farms; ③ both SGs and BESS; ④ synthetical responses (proposed mode). The planning results are listed in
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F009.jpg)
Fig. 9 Total cost and curtailment rate with different power penetration rates.
As can be observed from
In our planning model, the scenario technique is utilized to describe both the uncertainties of wind power and load demand by clustering the actual data set from 365 days into 4 typical days. However, the operation effectiveness that accommodates different levels of wind power and load fluctuation cannot be strictly guaranteed. In this vein, all planning schemes under different penetration rates are fixed as the input data. Then, the optimal operation simulation that considers fully actual data set is conducted. A higher possibility of wind curtailment and load shedding will be forced if the obtained schemes have violated certain operation constraints. Thus, we define violation rates of wind power and load demand as (96). Since all operation constraints in normal states can be totally satisfied through wind curtailment or load shedding, the normal operation problem is always feasible. However, the transient constraint of frequency nadir and ROCOF is possibly violated due to the limited frequency-support sources.
(96) |
where and count the number of days with wind curtailment and load shedding; and is the number of all simulation days.
In this subsection, we investigate the effects of unbalanced power on planning results.
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F010.jpg)
Fig. 10 Total cost and curtailment rate with different unbalanced power.
![](html/mpce/202104008/alternativeImage/2C1994C1-B98A-4d8b-8314-E3B2B614C1E1-F011.jpg)
Fig. 11 Frequency response power with different unbalanced power.
In this paper, a power-headroom constrained system frequency response model that incorporates thermal units, hydro units, wind farms, and BESS has been constructed to obtain synthetical frequency analysis. Afterwards, the multi-machine system has been transformed into a single generator by the parameter equivalence. Based on the DDPWL method, a linear coordination planning model of generation and battery energy storage has been presented to keep the frequency stability. Compared with the traditional generation planning problems, the proposed method guarantees adequate system inertia to limit ROCOF and supplies PFR to satisfy frequency nadir in future low-inertia power systems.
In our case studies, the proposed DDPWL method constructs linearized frequency constraints with high accuracy and enhanced computation efficiency. On this basis, we conduct a comparison of planning schemes in WFC and WOFC cases. Although the proposed method incurs more expansions as well as larger-scale online capacity, its effectiveness to satisfy frequency requirement is guaranteed. Moreover, the necessity of adopting virtual frequency responses for wind farms and BESSs is revealed. In such a way, more wind curtailment arises because of the sub-optimal de-loading mode, which addresses the importance of making a trade-off between wind power consumption and frequency-support. An excellent operation efficiency of the proposed model is also addressed through stochastic operation simulation. Furthermore, it can be concluded that an incremental unbalanced power incurs more installed devices since larger-scale frequency response power is required, while the effectiveness of our method to keep frequency stability is totally demonstrated.
In future work, efficient solution algorithms should be employed so that more representative days can be considered to describe uncertainty factors more precisely. Besides, more efficient control loops of frequency-support sources are expected to be applied for less conservative expansion schemes.
Appendix
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